Properties of a Determinant

Several structural properties make determinants easier to compute and manipulate, without ever multiplying everything out by hand.

• Multiplicativity: for two square matrices $A$ and $B$ of the same order, $|A|\,|B| = |AB|$ — the determinant of a product equals the product of the determinants.
• Scalar scaling: if $A = kB$ for matrices of order $n$, then $|A| = k^{n}|B|$; more generally, multiplying every element of a single row (or column) by a non-zero scalar $k$ multiplies the whole determinant by $k$, while multiplying all rows of an $n \times n$ matrix by $k$ multiplies the determinant by $k^{n}$.
• Transpose invariance: a matrix and its transpose always share the same determinant, $|A| = |A'|$.
• Row/column swap: interchanging any two rows (or any two columns) reverses the sign of the determinant.
• Repeated or proportional rows/columns: if two rows (or two columns) of a determinant are identical, or are proportional to one another, the determinant is exactly zero.
• Zero row/column: if every element of some row (or column) is zero, the determinant is zero.
• Linearity in a row: if the elements of one row (or column) can be split into a sum of two terms, the determinant splits accordingly into the sum of two separate determinants.
• Invariance under row operations: adding a scalar multiple of one row (or column) to another row (or column) — the operation $R_i \to R_i + kR_j$ — leaves the determinant completely unchanged. This is the single most useful property for hand-simplifying a determinant before expanding it. …